jmc

Since $f(x)$ is linear, $f(x)=mx+b$ for some constants $m$ and $b.$ From the equation $f(6)-f(2)=12,$ $$6m+b-(2m+b)=12.$$Then $4m=12,$ so $m=3.$ Therefore, $$f(12)-f(2)=12m+b-(2m+b)=10m=\boxed{30}.$$